The theory of Mean Field Games was invented roughly a decade ago simultaneously by Lasry-Lions on the one hand and Caines-Huang-Malhamé on the other hand. The aim of both groups was to study Nash equilibria of differential games with infinitely many players. In the first half of the talk, we will introduce some basic models and objects from the theory. A fundamental object — introduced by Lions in his lectures — that fully characterizes the equilibria, is the so-called master equation. This is an infinite dimensional nonlocal Hamilton-Jacobi equation set on the space of Borel probability measures endowed with a distance arising in the Monge-Kantorovich optimal transport problem. A central question in the theory is the global well-posedness of this equation in various settings. We will focus on master equations in absence of noise in the dynamics of the agents. Because of lack of smoothing effect (in the absence of diffusion), previously only a short time existence result of classical solutions (due to Gangbo-Swiech) was available. The highly nonlocal nature of the equation prevents us from developing a theory of viscosity solutions in this setting. In the second half of the talk — as part of an ongoing joint work with W. Gangbo — we present a possible approach to construct global in time classical solutions when the data satisfies a suitable convexity/monotonicity condition.